142 research outputs found
Lattice points in model domains of finite type in , II
We study the lattice point problem associated with a special class of
high-dimensional finite type domains via estimating the Fourier transforms of
corresponding indicator functions
An Optimal Transport View on Generalization
We derive upper bounds on the generalization error of learning algorithms
based on their \emph{algorithmic transport cost}: the expected Wasserstein
distance between the output hypothesis and the output hypothesis conditioned on
an input example. The bounds provide a novel approach to study the
generalization of learning algorithms from an optimal transport view and impose
less constraints on the loss function, such as sub-gaussian or bounded. We
further provide several upper bounds on the algorithmic transport cost in terms
of total variation distance, relative entropy (or KL-divergence), and VC
dimension, thus further bridging optimal transport theory and information
theory with statistical learning theory. Moreover, we also study different
conditions for loss functions under which the generalization error of a
learning algorithm can be upper bounded by different probability metrics
between distributions relating to the output hypothesis and/or the input data.
Finally, under our established framework, we analyze the generalization in deep
learning and conclude that the generalization error in deep neural networks
(DNNs) decreases exponentially to zero as the number of layers increases. Our
analyses of generalization error in deep learning mainly exploit the
hierarchical structure in DNNs and the contraction property of -divergence,
which may be of independent interest in analyzing other learning models with
hierarchical structure.Comment: 27 pages, 2 figures, 1 tabl
Theoretical Analysis of Adversarial Learning: A Minimax Approach
Here we propose a general theoretical method for analyzing the risk bound in
the presence of adversaries. Specifically, we try to fit the adversarial
learning problem into the minimax framework. We first show that the original
adversarial learning problem can be reduced to a minimax statistical learning
problem by introducing a transport map between distributions. Then, we prove a
new risk bound for this minimax problem in terms of covering numbers under a
weak version of Lipschitz condition. Our method can be applied to multi-class
classification problems and commonly used loss functions such as the hinge and
ramp losses. As some illustrative examples, we derive the adversarial risk
bounds for SVMs, deep neural networks, and PCA, and our bounds have two
data-dependent terms, which can be optimized for achieving adversarial
robustness.Comment: 27 pages, add some reference
An Information-Theoretic View for Deep Learning
Deep learning has transformed computer vision, natural language processing,
and speech recognition\cite{badrinarayanan2017segnet, dong2016image,
ren2017faster, ji20133d}. However, two critical questions remain obscure: (1)
why do deep neural networks generalize better than shallow networks; and (2)
does it always hold that a deeper network leads to better performance?
Specifically, letting be the number of convolutional and pooling layers in
a deep neural network, and be the size of the training sample, we derive an
upper bound on the expected generalization error for this network, i.e.,
\begin{eqnarray*}
\mathbb{E}[R(W)-R_S(W)] \leq
\exp{\left(-\frac{L}{2}\log{\frac{1}{\eta}}\right)}\sqrt{\frac{2\sigma^2}{n}I(S,W)
}
\end{eqnarray*} where is a constant depending on the loss
function, is a constant depending on the information loss for each
convolutional or pooling layer, and is the mutual information between
the training sample and the output hypothesis . This upper bound shows
that as the number of convolutional and pooling layers increases in the
network, the expected generalization error will decrease exponentially to zero.
Layers with strict information loss, such as the convolutional layers, reduce
the generalization error for the whole network; this answers the first
question. However, algorithms with zero expected generalization error does not
imply a small test error or . This is because
is large when the information for fitting the data is lost
as the number of layers increases. This suggests that the claim `the deeper the
better' is conditioned on a small training error or .
Finally, we show that deep learning satisfies a weak notion of stability and
the sample complexity of deep neural networks will decrease as increases.Comment: Add details in the proof of Theorem
Improving "Fast Iterative Shrinkage-Thresholding Algorithm": Faster, Smarter and Greedier
The "fast iterative shrinkage-thresholding algorithm", a.k.a. FISTA, is one
of the most well-known first-order optimisation scheme in the literature, as it
achieves the worst-case optimal convergence rate in terms of
objective function value. However, despite such an optimal theoretical
convergence rate, in practice the (local) oscillatory behaviour of FISTA often
damps its efficiency. Over the past years, various efforts are made in the
literature to improve the practical performance of FISTA, such as monotone
FISTA, restarting FISTA and backtracking strategies. In this paper, we propose
a simple yet effective modification to the original FISTA scheme which has two
advantages: it allows us to 1) prove the convergence of generated sequence; 2)
design a so-called "lazy-start" strategy which can up to an order faster than
the original scheme. Moreover, by exploring the properties of FISTA scheme, we
propose novel adaptive and greedy strategies which probes the limit of the
algorithm. The advantages of the proposed schemes are tested through problems
arising from inverse problem, machine learning and signal/image processing.Comment: correct proof of one lemm
A Novel Consensus-based Distributed Algorithm for Economic Dispatch Based on Local Estimation of Power Mismatch
This paper proposes a novel consensus-based distributed control algorithm for
solving the economic dispatch problem of distributed generators. A legacy
central controller can be eliminated in order to avoid a single point of
failure, relieve computational burden, maintain data privacy, and support
plug-and-play functionalities. The optimal economic dispatch is achieved by
allowing the iterative coordination of local agents (consumers and distributed
generators). As coordination information, the local estimation of power
mismatch is shared among distributed generators through communication networks
and does not contain any private information, ultimately contributing to a fair
electricity market. Additionally, the proposed distributed algorithm is
particularly designed for easy implementation and configuration of a large
number of agents in which the distributed decision making can be implemented in
a simple proportional-integral (PI) or integral (I) controller. In
MATLAB/Simulink simulation, the accuracy of the proposed distributed algorithm
is demonstrated in a 29-node system in comparison with the centralized
algorithm. Scalability and a fast convergence rate are also demonstrated in a
1400-node case study. Further, the experimental test demonstrates the practical
performance of the proposed distributed algorithm using the VOLTTRON platform
and a cluster of low-cost credit-card-size single-board PCs.Comment: 16 Pages, 13 figures Figures order and references are corrected
Experimental realization of quantum algorithms for linear system inspired by adiabatic quantum computing
Quantum adiabatic algorithm is of vital importance in quantum computation
field. It offers us an alternative approach to manipulate the system instead of
quantum gate model. Recently, an interesting work arXiv:1805.10549 indicated
that we can solve linear equation system via algorithm inspired by adiabatic
quantum computing. Here we demonstrate the algorithm and realize the solution
of 8-dimensional linear equations in a 4-qubit nuclear
magnetic resonance system. The result is by far the solution of
maximum-dimensional linear equation with a limited number of qubits in
experiments, which includes some ingenious simplifications. Our experiment
provides the new possibility of solving so many practical problems related to
linear equations systems and has the potential applications in designing the
future quantum algorithms
Quantum Pure State Tomography via Variational Hybrid Quantum-Classical Method
To obtain a complete description of a quantum system, one usually employs
standard quantum state tomography, which however requires exponential number of
measurements to perform and hence is impractical when the system's size grows
large. In this work, we introduce a self-learning tomographic scheme based on
the variational hybrid quantum-classical method. The key part of the scheme is
a learning procedure, in which we learn a control sequence capable of driving
the unknown target state coherently to a simple fiducial state, so that the
target state can be directly reconstructed by applying the control sequence
reversely. In this manner, the state tomography problem is converted to a
state-to-state transfer problem. To solve the latter problem, we use the
closed-loop learning control approach. Our scheme is further experimentally
tested using techniques of a 4-qubit nuclear magnetic resonance. {Experimental
results indicate that the proposed tomographic scheme can handle a broad class
of states including entangled states in quantum information, as well as
dynamical states of quantum many-body systems common to condensed matter
physics.Comment: 9 pages, 5 figures. To be published in Physical Review Applie
Experimental observation of information flow in the anti--symmetric system
The recently theoretical and experimental researches related to
-symmetric system have attracted unprecedented attention because
of various novel features and potentials in extending canonical quantum
mechanics. However, as the counterpart of -symmetry, there are
only a few researches on anti--symmetry. Here, we propose an
algorithm for simulating the universal anti--symmetric system
with quantum circuit. Utilizing the protocols, an oscillation of information
flow is observed for the first time in our Nuclear Magnetic Resonance quantum
simulator. We will show that information will recover from the environment
completely when the anti--symmetry is broken, whereas no
information can be retrieved in the symmetry-unbroken phase. Our work opens the
gate for practical quantum simulation and experimental investigation of
universal anti--symmetric system in quantum computer
AlignShift: Bridging the Gap of Imaging Thickness in 3D Anisotropic Volumes
This paper addresses a fundamental challenge in 3D medical image processing:
how to deal with imaging thickness. For anisotropic medical volumes, there is a
significant performance gap between thin-slice (mostly 1mm) and thick-slice
(mostly 5mm) volumes. Prior arts tend to use 3D approaches for the thin-slice
and 2D approaches for the thick-slice, respectively. We aim at a unified
approach for both thin- and thick-slice medical volumes. Inspired by recent
advances in video analysis, we propose AlignShift, a novel parameter-free
operator to convert theoretically any 2D pretrained network into
thickness-aware 3D network. Remarkably, the converted networks behave like 3D
for the thin-slice, nevertheless degenerate to 2D for the thick-slice
adaptively. The unified thickness-aware representation learning is achieved by
shifting and fusing aligned "virtual slices" as per the input imaging
thickness. Extensive experiments on public large-scale DeepLesion benchmark,
consisting of 32K lesions for universal lesion detection, validate the
effectiveness of our method, which outperforms previous state of the art by
considerable margins without whistles and bells. More importantly, to our
knowledge, this is the first method that bridges the performance gap between
thin- and thick-slice volumes by a unified framework. To improve research
reproducibility, our code in PyTorch is open source at
https://github.com/M3DV/AlignShift.Comment: MICCAI 2020 (early accepted). Camera ready version. Code is available
at https://github.com/M3DV/AlignShif
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